A $p$-adic local monodromy theorem
We produce a canonical filtration for locally free sheaves on an open $p$-adic annulus equipped with a Frobenius structure. Using this filtration, we deduce a conjecture of Crew on $p$-adic...
View ArticleThe subconvexity problem for Rankin-Selberg $L$-functions and...
In this paper we solve the subconvexity problem for Rankin-Selberg $L$-functions $L(f\otimes g,s)$ where $f$ and $g$ are two cuspidal automorphic forms over $\mathbb{Q}$, $g$ being fixed and $f$ having...
View ArticleDetermination of the algebraic relations among special $\Gamma$-values in...
We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual $t$-motives, we find that all algebraic relations among special values of the geometric...
View ArticleRemovability of point singularities of Willmore surfaces
We investigate point singularities of Willmore surfaces, which for example appear as blowups of the Willmore flow near singularities, and prove that closed Willmore surfaces with one unit density point...
View ArticleGrothendieck’s problems concerning profinite completions and representations...
In 1970 Alexander Grothendieck [6] posed the following problem: let $\Gamma_1$ and $\Gamma_2$ be finitely presented, residually finite groups, and let $u:\Gamma_1\to\Gamma_2$ be a homomorphism such...
View ArticleRobust transitive singular sets for 3-flows are partially hyperbolic...
Inspired by Lorenz’ remarkable chaotic flow, we describe in this paper the structure of all $C^1$ robust transitive sets with singularities for flows on closed $3$-manifolds: they are partially...
View ArticleCover times for Brownian motion and random walks in two dimensions
Let $\mathcal{T}(x,\varepsilon)$ denote the first hitting time of the disc of radius $\varepsilon$ centered at $x$ for Brownian motion on the two dimensional torus $\mathbb{T}^2$. We prove that...
View ArticleGeometry of the uniform spanning forest: Transitions in dimensions 4, 8, 12,…
The uniform spanning forest (USF) in $\mathbb{Z}^d$ is the weak limit of random, uniformly chosen, spanning trees in $[-n,n]^d$. Pemantle [11] proved that the USF consists a.s. of a single tree if and...
View ArticleClassification of local conformal nets. Case $c < 1$
We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge $c$ less than 1. The irreducible ones are in bijective correspondence with the pairs...
View ArticleUnique decomposition of tensor products of irreducible representations of...
We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero determines the individual constituents uniquely. This is...
View ArticleOn integral points on surfaces
We study the integral points on surfaces by means of a new method, relying on the Schmidt Subspace Theorem. This method was recently introduced in [CZ] for the case of curves, leading to a new proof of...
View ArticleOn the periods of motives with complex multiplication and a conjecture of...
We prove that the existence of an automorphism of finite order on a $\overline{\mathbb{Q}}$-variety $X$ implies the existence of algebraic linear relations between the logarithm of certain periods of...
View ArticleKloosterman identities over quadratic extension
We prove an identity of Kloosterman integrals which is the fundamental lemma of a relative trace formula for the general linear group in $n$ variables.
View ArticlePRIMES is in P
We present an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite.
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